Calculus C Notes - Chapters 7 and 8

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Chapter 7
Techniques of Integration
7-1 Integration by Parts
7-2 Trigonometric Integrals
7-3 Trigonometric Substitution
7-4 Integration by Partial Fractions
7-6 Integration by Tables
L’Hôpital’s Rule
7-8 Improper Integrals
The following notes are for the Calculus C (SDSU Math 151)
classes I teach at Torrey Pines High School. I wrote and
modified these notes over several semesters. The
explanations are my own; however, I borrowed several
examples and diagrams from the textbooks* my classes used
while I taught the course. Over time, I have changed some
examples and have forgotten which ones came from which
sources. Also, I have chosen to keep the notes in my own
handwriting rather than type to maintain their informality
and to avoid the tedious task of typing so many formulas,
equations, and diagrams. These notes are free for use by my
current and former students. If other calculus students and
teachers find these notes useful, I would be happy to know
that my work was helpful. - Abby Brown
SDUHSD Calculus II/C
SDSU Math 151
Abby Brown
www.abbymath.com
San Diego, CA
*Calculus: Early Transcendentals, 6th & 4th editions, James Stewart, ©2007 & 1999
Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.
(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)
*Calculus, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards, ©1994
D. C. Heath and Company, ISBN 0-669-35335-3.
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Chapter 8
Further Applications
of Integration
8-1 Arc Length
8-2 Area of a Surface of Revolution
SDUHSD Calculus II/C
SDSU Math 151
Abby Brown
www.abbymath.com
San Diego, CA
The following notes are for the Calculus C (SDSU Math 151)
classes I teach at Torrey Pines High School. I wrote and
modified these notes over several semesters. The
explanations are my own; however, I borrowed several
examples and diagrams from the textbooks* my classes used
while I taught the course. Over time, I have changed some
examples and have forgotten which ones came from which
sources. Also, I have chosen to keep the notes in my own
handwriting rather than type to maintain their informality
and to avoid the tedious task of typing so many formulas,
equations, and diagrams. These notes are free for use by my
current and former students. If other calculus students and
teachers find these notes useful, I would be happy to know
that my work was helpful. - Abby Brown
*Calculus: Early Transcendentals, 6th & 4th editions, James Stewart, ©2007 & 1999
Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.
(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)
*Calculus, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards, ©1994
D. C. Heath and Company, ISBN 0-669-35335-3.
Name: ___________________________________
www.abbymath.com - Ch. 7 & 8
Page 17 of 20
lower-case
Note that this is just a general sketch of the proof that depends on the Mean Value Theorem.
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upper-case
Note that this is a general sketch and not a formal proof.
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Calculus C – Exam #1 Review
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Chapter 7 – Techniques of Integration
Larson 7.1 Review of Basic Techniques
7.1
Integration by Parts
7.2
Trigonometric Integrals
7.3
Trigonometric Substitution
7.4
Integration of Rational Functions by Partial Fractions
7.5
Strategy for Integration
7.6
Integration Using Tables
Larson 7.7 L’Hôpital’s Rule (Remember: Only 0/0 or ±4/±4 forms)
7.8
Improper Integrals (Don’t Forget: Write lim , etc. where needed.)
b→ ∞
Chapter 8 – Further Applications of Integration
8.1
Arc Length
8.2
Area of a Surface of Revolution
Integration: How am I supposed to know which method to use?
0th
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
Multiply out powers to get several small integrals
Look for a possible u-substitution
Pick some “inside” function
Is the derivative of your u available to become part of du?
Can the integrand be adjusted slightly so that a basic u-substitution will work?
Multiply by a constant of by “1” with variables
Adding “0” by (+) and (-) a constant
Can the integral be split?
Simple numerator split
Partial Fractions Decomposition
Is the integrand an improper rational expression?
Use long division to split
Is it inverse trig.? Or Trigonometric Substitution?
x2 + a2
arctan, arcsin, arcsec
x
Draw a triangle (3 types)
θ
a
Can I complete the square to make it look like inverse trig.?
Don’t forget to also subtract anything new you add
Would trig. identities help?
cos2 x = (1+ cos 2x)/2
cos2 x + sin2 x = 1
2
2
cos x - sin x = cos 2x
sin2 x = (1- cos 2x)/2
2 sin x cos x = sin 2 x
1 + tan2 x = sec2 x
Try integration by parts
u=
dv =____dx
du =____dx v =
Y uv - Iv du
Refer to integral tables
Find a match, adjust using u-substitution
Use an electronic, algebraic solver
If integral is definite, use numerical methods
Left- and Right-Hand Rectangles
Midpoint Rectangles
Trapezoidal Rule
Simpson’s Rule